calculations were carried out for SFs of ZPVE corrections [45, 46]. In addition, SFs
for a few dozen of common computational levels along with the calculated
uncertainties were reported in both works [44, 45]. High uncertainties of scaling
factors were also confirmed by Cordeiro et al. [47] who carried out calculations for
X3LYP functional. The considerations reported above opened a debate on the
uncertainties of SFs [48–50]. High uncertainties of SFs put US procedure in a little
bad light. However, as stated in the pioneering work [44] “…until the underlying
probability distributions have been characterized, it is not possible to obtain
probabilistic uncertainty intervals. Improved methods for classifying vibrational
frequencies will probably lead to distributions that are more nearly normal and to
smaller uncertainties.” It seems at the moment that the SFs uncertainty problem has
not been fully solved yet.
2.3.3 Wavenumber Linear Scaling
Another single-parameter scaling procedure was devised by Yoshida and
co-workers in 2000 [51]. It is called wavenumber linear scaling (WLS). To our best
knowledge, only two methodological, frequently cited works from early 2000s
appeared in the literature [51, 52] so far.
The authors considered 205 organic and inorganic molecules, for which they
considered as many as 1729 fundamentals [51]. Then, after calculating harmonic
frequencies at the B3LYP/6-311+G** computational level they noticed that the ratio
f ¼
m
expt
m h , called wavenumber scale factor, is typically within 1.00 ± 0.05 range
(although larger deviations were also observed) and decreases linearly with increasing
frequency. In addition, they noticed that at high frequencies the spread in the factor is
fairly low and increases along with the decreasing wavenumber. After rejecting 28
modes from the set, for which deviations of f were higher than 10%, i.e., using 1701
fundamentals, they obtained the relation for the frequency-dependent optimal SF of
the form
f
opt m
h
À Á ¼ 1:0087 9
ð Þ À 0:0000163 6
ð Þm
h
ð2:44Þ
where m
h is expressed in [cm
−1 ] and the value in parentheses are uncertainties at the
last significant figure. In fact, similar relation was provided at the same time [53],
but it seems to be not as general as Yoshida’s one, since only a few molecules and
merely 139 fundamentals were considered.
The idea of the WLS was extended by Yoshida et al. two years later [52]. The
authors performed additional calculations on 164 organic and inorganic compounds
using the same as before computational level and scaled them according to
Eq. (2.44). The average deviation from the experimental values was 3.4% for 1223
vibrational modes. In addition, they examined a set of 224 diatomic molecules and
ions, for which experimental harmonic frequencies were available. After calculating
m
expt;h
m h , they observed that in more than 90% cases the ratio was 1.0 ± 0.1 and
72
O. Bąk and P. Borowski
for a few dozen of common computational levels along with the calculated
uncertainties were reported in both works [44, 45]. High uncertainties of scaling
factors were also confirmed by Cordeiro et al. [47] who carried out calculations for
X3LYP functional. The considerations reported above opened a debate on the
uncertainties of SFs [48–50]. High uncertainties of SFs put US procedure in a little
bad light. However, as stated in the pioneering work [44] “…until the underlying
probability distributions have been characterized, it is not possible to obtain
probabilistic uncertainty intervals. Improved methods for classifying vibrational
frequencies will probably lead to distributions that are more nearly normal and to
smaller uncertainties.” It seems at the moment that the SFs uncertainty problem has
not been fully solved yet.
2.3.3 Wavenumber Linear Scaling
Another single-parameter scaling procedure was devised by Yoshida and
co-workers in 2000 [51]. It is called wavenumber linear scaling (WLS). To our best
knowledge, only two methodological, frequently cited works from early 2000s
appeared in the literature [51, 52] so far.
The authors considered 205 organic and inorganic molecules, for which they
considered as many as 1729 fundamentals [51]. Then, after calculating harmonic
frequencies at the B3LYP/6-311+G** computational level they noticed that the ratio
f ¼
m
expt
m h , called wavenumber scale factor, is typically within 1.00 ± 0.05 range
(although larger deviations were also observed) and decreases linearly with increasing
frequency. In addition, they noticed that at high frequencies the spread in the factor is
fairly low and increases along with the decreasing wavenumber. After rejecting 28
modes from the set, for which deviations of f were higher than 10%, i.e., using 1701
fundamentals, they obtained the relation for the frequency-dependent optimal SF of
the form
f
opt m
h
À Á ¼ 1:0087 9
ð Þ À 0:0000163 6
ð Þm
h
ð2:44Þ
where m
h is expressed in [cm
−1 ] and the value in parentheses are uncertainties at the
last significant figure. In fact, similar relation was provided at the same time [53],
but it seems to be not as general as Yoshida’s one, since only a few molecules and
merely 139 fundamentals were considered.
The idea of the WLS was extended by Yoshida et al. two years later [52]. The
authors performed additional calculations on 164 organic and inorganic compounds
using the same as before computational level and scaled them according to
Eq. (2.44). The average deviation from the experimental values was 3.4% for 1223
vibrational modes. In addition, they examined a set of 224 diatomic molecules and
ions, for which experimental harmonic frequencies were available. After calculating
m
expt;h
m h , they observed that in more than 90% cases the ratio was 1.0 ± 0.1 and
72
O. Bąk and P. Borowski